Question
Compare the graphs of f(x)=x+1−−−−√−2
and g(x)=x+1−−−−√^3−2
. Identify which of the following features is true about the graphs.
(1 point)
Responses
The graphs have the same x-intercept.
The graphs have the same x -intercept.
The graphs have the same domains.
The graphs have the same domains.
The graphs have the same ranges.
The graphs have the same ranges.
The graphs have the same y-intercept.
The graphs have the same y -intercept.
and g(x)=x+1−−−−√^3−2
. Identify which of the following features is true about the graphs.
(1 point)
Responses
The graphs have the same x-intercept.
The graphs have the same x -intercept.
The graphs have the same domains.
The graphs have the same domains.
The graphs have the same ranges.
The graphs have the same ranges.
The graphs have the same y-intercept.
The graphs have the same y -intercept.
Answers
GPT-4o mini
To compare the graphs of the functions \( f(x) = \sqrt{x + 1} - 2 \) and \( g(x) = (\sqrt{x + 1})^3 - 2 \), we will analyze their features, including x-intercepts, domains, ranges, and y-intercepts.
### Domain
1. **For \( f(x) \)**: The expression \( \sqrt{x + 1} \) is defined when \( x + 1 \geq 0 \), which simplifies to \( x \geq -1 \). Therefore, the domain of \( f(x) \) is \( [-1, \infty) \).
2. **For \( g(x) \)**: The expression \( \sqrt{x + 1} \) is also defined under the same condition \( x + 1 \geq 0 \), leading to the same domain of \( g(x) \) as \( [-1, \infty) \).
### X-Intercept
To find the x-intercepts, we set each function to zero:
1. **For \( f(x) = 0 \)**:
\[
\sqrt{x + 1} - 2 = 0 \implies \sqrt{x + 1} = 2 \implies x + 1 = 4 \implies x = 3
\]
2. **For \( g(x) = 0 \)**:
\[
(\sqrt{x + 1})^3 - 2 = 0 \implies (\sqrt{x + 1})^3 = 2 \implies \sqrt{x + 1} = \sqrt[3]{2} \implies x + 1 = (\sqrt[3]{2})^2 \implies x = \sqrt[3]{2}^2 - 1
\]
The value of \( x \) for the x-intercept of \( g(x) \) is different from \( 3 \) (since \( \sqrt[3]{2} \) is not equal to \( 2 \)).
### Y-Intercept
To find the y-intercepts, substitute \( x = 0 \):
1. **For \( f(0) \)**:
\[
f(0) = \sqrt{0 + 1} - 2 = 1 - 2 = -1
\]
2. **For \( g(0) \)**:
\[
g(0) = (\sqrt{0 + 1})^3 - 2 = 1^3 - 2 = 1 - 2 = -1
\]
Both functions have the same y-intercept at \( (0, -1) \).
### Range
1. **For \( f(x) \)**: The minimum value occurs when \( x = -1 \):
\[
f(-1) = \sqrt{-1 + 1} - 2 = 0 - 2 = -2.
\]
As \( x \) increases, \( f(x) \) approaches infinity. Therefore, the range is \( [-2, \infty) \).
2. **For \( g(x) \)**:
- The minimum value occurs at \( x = -1 \):
\[
g(-1) = (\sqrt{-1 + 1})^3 - 2 = 0 - 2 = -2.
\]
As \( x \) increases, \( g(x) \) also approaches infinity but at a different rate due to the cubic transformation. Therefore, the range is also \( [-2, \infty) \).
### Conclusion
Now we can summarize our findings:
- **The graphs have the same x-intercept**: False
- **The graphs have the same domains**: True
- **The graphs have the same ranges**: True
- **The graphs have the same y-intercept**: True
Since all features indicated that the graphs have the same domains, ranges, and y-intercepts, the true statements about the graphs are:
- The graphs have the same domains.
- The graphs have the same ranges.
- The graphs have the same y-intercept.
However, since the question requested one feature, and if choosing only one is necessary, the statement "The graphs have the same domains." is correct.
### Domain
1. **For \( f(x) \)**: The expression \( \sqrt{x + 1} \) is defined when \( x + 1 \geq 0 \), which simplifies to \( x \geq -1 \). Therefore, the domain of \( f(x) \) is \( [-1, \infty) \).
2. **For \( g(x) \)**: The expression \( \sqrt{x + 1} \) is also defined under the same condition \( x + 1 \geq 0 \), leading to the same domain of \( g(x) \) as \( [-1, \infty) \).
### X-Intercept
To find the x-intercepts, we set each function to zero:
1. **For \( f(x) = 0 \)**:
\[
\sqrt{x + 1} - 2 = 0 \implies \sqrt{x + 1} = 2 \implies x + 1 = 4 \implies x = 3
\]
2. **For \( g(x) = 0 \)**:
\[
(\sqrt{x + 1})^3 - 2 = 0 \implies (\sqrt{x + 1})^3 = 2 \implies \sqrt{x + 1} = \sqrt[3]{2} \implies x + 1 = (\sqrt[3]{2})^2 \implies x = \sqrt[3]{2}^2 - 1
\]
The value of \( x \) for the x-intercept of \( g(x) \) is different from \( 3 \) (since \( \sqrt[3]{2} \) is not equal to \( 2 \)).
### Y-Intercept
To find the y-intercepts, substitute \( x = 0 \):
1. **For \( f(0) \)**:
\[
f(0) = \sqrt{0 + 1} - 2 = 1 - 2 = -1
\]
2. **For \( g(0) \)**:
\[
g(0) = (\sqrt{0 + 1})^3 - 2 = 1^3 - 2 = 1 - 2 = -1
\]
Both functions have the same y-intercept at \( (0, -1) \).
### Range
1. **For \( f(x) \)**: The minimum value occurs when \( x = -1 \):
\[
f(-1) = \sqrt{-1 + 1} - 2 = 0 - 2 = -2.
\]
As \( x \) increases, \( f(x) \) approaches infinity. Therefore, the range is \( [-2, \infty) \).
2. **For \( g(x) \)**:
- The minimum value occurs at \( x = -1 \):
\[
g(-1) = (\sqrt{-1 + 1})^3 - 2 = 0 - 2 = -2.
\]
As \( x \) increases, \( g(x) \) also approaches infinity but at a different rate due to the cubic transformation. Therefore, the range is also \( [-2, \infty) \).
### Conclusion
Now we can summarize our findings:
- **The graphs have the same x-intercept**: False
- **The graphs have the same domains**: True
- **The graphs have the same ranges**: True
- **The graphs have the same y-intercept**: True
Since all features indicated that the graphs have the same domains, ranges, and y-intercepts, the true statements about the graphs are:
- The graphs have the same domains.
- The graphs have the same ranges.
- The graphs have the same y-intercept.
However, since the question requested one feature, and if choosing only one is necessary, the statement "The graphs have the same domains." is correct.